We consider packet scheduling for the downlink in a wireless network, where each packet's service preferences are captured by a utility function that depends on the packet's delay. The goal is to schedule packet transmissions to maximize the total utility. We examine a simple gradient-based scheduling algorithm, theUR-rule, which is a type of generalized cµ-rule (Gcµ) that takes into account both a user's channel condition and derived utility. We study the performance of this scheduling rule

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... a draining problem. We formulate a "large system" fluid model for this draining problem where the number of packets increases while the packet-size decreases to zero, and give a complete characterization of the behavior of theUR scheduling rule in this limiting regime. We then give an optimal control formulation for finding the optimal scheduling policy for the fluid draining model. Using Pontryagin's minimum principle, we show that, when the user rates are chosen from a TDM-type of capacity region, theUR rule is in fact optimal in many cases. Finally, we consider non-TDM capacity regions and show that here theUR rule is optimal only in special cases. 1 A utility U that is a function of delay is equivalent to a delay cost of −U.